# Properties

 Label 2205.4.a.w Level $2205$ Weight $4$ Character orbit 2205.a Self dual yes Analytic conductor $130.099$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2205.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$130.099211563$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{11})$$ Defining polynomial: $$x^{2} - 11$$ x^2 - 11 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 245) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + ( - 2 \beta + 4) q^{4} - 5 q^{5} + ( - 2 \beta - 18) q^{8}+O(q^{10})$$ q + (b - 1) * q^2 + (-2*b + 4) * q^4 - 5 * q^5 + (-2*b - 18) * q^8 $$q + (\beta - 1) q^{2} + ( - 2 \beta + 4) q^{4} - 5 q^{5} + ( - 2 \beta - 18) q^{8} + ( - 5 \beta + 5) q^{10} + ( - 4 \beta - 33) q^{11} + ( - 20 \beta + 5) q^{13} - 36 q^{16} + ( - 20 \beta - 35) q^{17} + ( - 20 \beta + 70) q^{19} + (10 \beta - 20) q^{20} + ( - 29 \beta - 11) q^{22} + ( - 28 \beta + 8) q^{23} + 25 q^{25} + (25 \beta - 225) q^{26} + ( - 48 \beta + 129) q^{29} + (60 \beta - 10) q^{31} + ( - 20 \beta + 180) q^{32} + ( - 15 \beta - 185) q^{34} + ( - 44 \beta + 164) q^{37} + (90 \beta - 290) q^{38} + (10 \beta + 90) q^{40} + ( - 100 \beta - 150) q^{41} + ( - 12 \beta - 58) q^{43} + (50 \beta - 44) q^{44} + (36 \beta - 316) q^{46} + ( - 40 \beta + 15) q^{47} + (25 \beta - 25) q^{50} + ( - 90 \beta + 460) q^{52} + ( - 120 \beta - 270) q^{53} + (20 \beta + 165) q^{55} + (177 \beta - 657) q^{58} + (40 \beta - 190) q^{59} + (60 \beta + 540) q^{61} + ( - 70 \beta + 670) q^{62} + (200 \beta - 112) q^{64} + (100 \beta - 25) q^{65} + (96 \beta + 234) q^{67} + ( - 10 \beta + 300) q^{68} + (64 \beta + 528) q^{71} + (200 \beta - 430) q^{73} + (208 \beta - 648) q^{74} + ( - 220 \beta + 720) q^{76} + ( - 348 \beta + 79) q^{79} + 180 q^{80} + ( - 50 \beta - 950) q^{82} + ( - 200 \beta - 20) q^{83} + (100 \beta + 175) q^{85} + ( - 46 \beta - 74) q^{86} + (138 \beta + 682) q^{88} + (380 \beta + 120) q^{89} + ( - 128 \beta + 648) q^{92} + (55 \beta - 455) q^{94} + (100 \beta - 350) q^{95} + ( - 180 \beta + 815) q^{97}+O(q^{100})$$ q + (b - 1) * q^2 + (-2*b + 4) * q^4 - 5 * q^5 + (-2*b - 18) * q^8 + (-5*b + 5) * q^10 + (-4*b - 33) * q^11 + (-20*b + 5) * q^13 - 36 * q^16 + (-20*b - 35) * q^17 + (-20*b + 70) * q^19 + (10*b - 20) * q^20 + (-29*b - 11) * q^22 + (-28*b + 8) * q^23 + 25 * q^25 + (25*b - 225) * q^26 + (-48*b + 129) * q^29 + (60*b - 10) * q^31 + (-20*b + 180) * q^32 + (-15*b - 185) * q^34 + (-44*b + 164) * q^37 + (90*b - 290) * q^38 + (10*b + 90) * q^40 + (-100*b - 150) * q^41 + (-12*b - 58) * q^43 + (50*b - 44) * q^44 + (36*b - 316) * q^46 + (-40*b + 15) * q^47 + (25*b - 25) * q^50 + (-90*b + 460) * q^52 + (-120*b - 270) * q^53 + (20*b + 165) * q^55 + (177*b - 657) * q^58 + (40*b - 190) * q^59 + (60*b + 540) * q^61 + (-70*b + 670) * q^62 + (200*b - 112) * q^64 + (100*b - 25) * q^65 + (96*b + 234) * q^67 + (-10*b + 300) * q^68 + (64*b + 528) * q^71 + (200*b - 430) * q^73 + (208*b - 648) * q^74 + (-220*b + 720) * q^76 + (-348*b + 79) * q^79 + 180 * q^80 + (-50*b - 950) * q^82 + (-200*b - 20) * q^83 + (100*b + 175) * q^85 + (-46*b - 74) * q^86 + (138*b + 682) * q^88 + (380*b + 120) * q^89 + (-128*b + 648) * q^92 + (55*b - 455) * q^94 + (100*b - 350) * q^95 + (-180*b + 815) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 8 q^{4} - 10 q^{5} - 36 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 8 * q^4 - 10 * q^5 - 36 * q^8 $$2 q - 2 q^{2} + 8 q^{4} - 10 q^{5} - 36 q^{8} + 10 q^{10} - 66 q^{11} + 10 q^{13} - 72 q^{16} - 70 q^{17} + 140 q^{19} - 40 q^{20} - 22 q^{22} + 16 q^{23} + 50 q^{25} - 450 q^{26} + 258 q^{29} - 20 q^{31} + 360 q^{32} - 370 q^{34} + 328 q^{37} - 580 q^{38} + 180 q^{40} - 300 q^{41} - 116 q^{43} - 88 q^{44} - 632 q^{46} + 30 q^{47} - 50 q^{50} + 920 q^{52} - 540 q^{53} + 330 q^{55} - 1314 q^{58} - 380 q^{59} + 1080 q^{61} + 1340 q^{62} - 224 q^{64} - 50 q^{65} + 468 q^{67} + 600 q^{68} + 1056 q^{71} - 860 q^{73} - 1296 q^{74} + 1440 q^{76} + 158 q^{79} + 360 q^{80} - 1900 q^{82} - 40 q^{83} + 350 q^{85} - 148 q^{86} + 1364 q^{88} + 240 q^{89} + 1296 q^{92} - 910 q^{94} - 700 q^{95} + 1630 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 + 8 * q^4 - 10 * q^5 - 36 * q^8 + 10 * q^10 - 66 * q^11 + 10 * q^13 - 72 * q^16 - 70 * q^17 + 140 * q^19 - 40 * q^20 - 22 * q^22 + 16 * q^23 + 50 * q^25 - 450 * q^26 + 258 * q^29 - 20 * q^31 + 360 * q^32 - 370 * q^34 + 328 * q^37 - 580 * q^38 + 180 * q^40 - 300 * q^41 - 116 * q^43 - 88 * q^44 - 632 * q^46 + 30 * q^47 - 50 * q^50 + 920 * q^52 - 540 * q^53 + 330 * q^55 - 1314 * q^58 - 380 * q^59 + 1080 * q^61 + 1340 * q^62 - 224 * q^64 - 50 * q^65 + 468 * q^67 + 600 * q^68 + 1056 * q^71 - 860 * q^73 - 1296 * q^74 + 1440 * q^76 + 158 * q^79 + 360 * q^80 - 1900 * q^82 - 40 * q^83 + 350 * q^85 - 148 * q^86 + 1364 * q^88 + 240 * q^89 + 1296 * q^92 - 910 * q^94 - 700 * q^95 + 1630 * q^97

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.31662 3.31662
−4.31662 0 10.6332 −5.00000 0 0 −11.3668 0 21.5831
1.2 2.31662 0 −2.63325 −5.00000 0 0 −24.6332 0 −11.5831
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2205.4.a.w 2
3.b odd 2 1 245.4.a.j yes 2
7.b odd 2 1 2205.4.a.x 2
15.d odd 2 1 1225.4.a.p 2
21.c even 2 1 245.4.a.i 2
21.g even 6 2 245.4.e.k 4
21.h odd 6 2 245.4.e.j 4
105.g even 2 1 1225.4.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.i 2 21.c even 2 1
245.4.a.j yes 2 3.b odd 2 1
245.4.e.j 4 21.h odd 6 2
245.4.e.k 4 21.g even 6 2
1225.4.a.p 2 15.d odd 2 1
1225.4.a.q 2 105.g even 2 1
2205.4.a.w 2 1.a even 1 1 trivial
2205.4.a.x 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2205))$$:

 $$T_{2}^{2} + 2T_{2} - 10$$ T2^2 + 2*T2 - 10 $$T_{11}^{2} + 66T_{11} + 913$$ T11^2 + 66*T11 + 913 $$T_{13}^{2} - 10T_{13} - 4375$$ T13^2 - 10*T13 - 4375 $$T_{17}^{2} + 70T_{17} - 3175$$ T17^2 + 70*T17 - 3175

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 10$$
$3$ $$T^{2}$$
$5$ $$(T + 5)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 66T + 913$$
$13$ $$T^{2} - 10T - 4375$$
$17$ $$T^{2} + 70T - 3175$$
$19$ $$T^{2} - 140T + 500$$
$23$ $$T^{2} - 16T - 8560$$
$29$ $$T^{2} - 258T - 8703$$
$31$ $$T^{2} + 20T - 39500$$
$37$ $$T^{2} - 328T + 5600$$
$41$ $$T^{2} + 300T - 87500$$
$43$ $$T^{2} + 116T + 1780$$
$47$ $$T^{2} - 30T - 17375$$
$53$ $$T^{2} + 540T - 85500$$
$59$ $$T^{2} + 380T + 18500$$
$61$ $$T^{2} - 1080 T + 252000$$
$67$ $$T^{2} - 468T - 46620$$
$71$ $$T^{2} - 1056 T + 233728$$
$73$ $$T^{2} + 860T - 255100$$
$79$ $$T^{2} - 158 T - 1325903$$
$83$ $$T^{2} + 40T - 439600$$
$89$ $$T^{2} - 240 T - 1574000$$
$97$ $$T^{2} - 1630 T + 307825$$